Para Blaschke isoparametric spacelike hypersurfaces in Lorentzian space forms

TL;DR

This paper classifies para-Blaschke isoparametric spacelike hypersurfaces in Lorentzian space forms, focusing on those with constant eigenvalues of a combined conformal tensor, under conformal transformations.

Contribution

It provides a classification of para-Blaschke isoparametric spacelike hypersurfaces in Lorentzian space forms, extending the understanding of conformal invariants in Lorentzian geometry.

Findings

Classification of para-Blaschke isoparametric hypersurfaces achieved

Characterization of hypersurfaces with constant eigenvalues of the para-Blaschke tensor

Identification of geometric conditions under conformal transformations

Abstract

Let $M^n$ be an $n$-dimensional umbilic-free hypersurface in the $(n+1)$-dimensional Lorentzian space form $M^{n+1}_1(c)$. Three basic invariants of $M^n$ under the conformal transformation group of $M^{n+1}_1(c)$ are a $1$-form $C$, called conformal $1$-form, a symmetric $(0,2)$ tensor $B$, called conformal second fundamental form, and a symmetric $(0,2)$ tensor $A$, called Blaschke tensor. The so-called para-Blaschke tensor $D^{\lambda}=A+\lambda B$, the linear combination of $A$ and $B$, is still a symmetric $(0,2)$ tensor. A spacelike hypersurface is called a para-Blaschke isoparametric spacelike hypersurface, if the conform $1$-form vanishes and the eigenvalues of the para-Blaschke tensor are constant. In this paper, we classify the para-Blaschke isoparametric spacelike hypersurfaces under the conformal group of $M^{n+1}_1(c)$.

Equations

$$⟨X,Y{⟩}_s=-\sum _{i=1}^s{x}_i{y}_i+\sum _{j=s+1}^{n+2}{x}_j{y}_j.$$

$$\begin{array}{r}{\mathbb{S}}^{n+1}(a)=\{x\in {\mathbb{R}}^{n+2}|x\cdot x=a^2\},{\mathbb{H}}^{n+1}(-a)=\{x\in {\mathbb{R}}_1^{n+2}|⟨x,x{⟩}_1=-a^2\},\\ {\mathbb{S}}_1^{n+1}(a)=\{x\in {\mathbb{R}}_1^{n+2}|⟨x,x{⟩}_1=a^2\},{\mathbb{H}}_1^{n+1}(-a)=\{x\in {\mathbb{R}}_2^{n+2}|⟨x,x{⟩}_2=-a^2\}.\end{array}$$

$$x(u^{\prime },u^{\prime \prime },t,u^{\prime \prime \prime })=(t{u}^{\prime },t{u}^{\prime \prime },u^{\prime \prime \prime }),$$

$$x:{\mathbb{H}}^q(-\sqrt{a^2-1})\times {\mathbb{S}}^p(a)\times {\mathbb{R}}^{+}\times {\mathbb{R}}^{n-p-q-1}\to {\mathbb{R}}_1^{n+1},$$

$$C^{n+2}=\{X\in {\mathbb{R}}_2^{n+3}|⟨X,X{⟩}_2=0,X\ne 0\},$$

$${\mathbb{Q}}_1^{n+1}=\{[X]\in \mathbb{R}{P}^{n+2}|⟨X,X{⟩}_2=0\}.$$

$$T([X])=[XT],X\in C^{n+2},T\in O(n+3,2).$$

$$[h]=\{e^{\tau }h|\tau \in C^{\infty }({\mathbb{Q}}_1^{n+1})\}$$

$$\begin{array}[]{l}\sigma_{0}:\mathbb{R}^{n+1}_{1}\rightarrow{\mathbb{Q}}^{n+1}_{1}\backslash P,~{}~{}~{}~{}\quad u\mapsto[(\frac{<u,u>_{1}+1}{2},u,\frac{<u,u>_{1}-1}{2})],\\ \sigma_{1}:\mathbb{S}^{n+1}_{1}(1)\rightarrow{\mathbb{Q}}^{n+1}_{1}\backslash P_{+},~{}~{}\quad u\mapsto[(1,u)],\\ \sigma_{-1}:\mathbb{H}^{n+1}_{1}(-1)\rightarrow{\mathbb{Q}}^{n+1}_{1}\backslash P_{-},~{}\quad u\mapsto[(u,1)].\\ \end{array}$$

$$Y:M\to C^{n+2}$$

$$N=-\frac{1}{n}\Delta Y-\frac{1}{2n^2}⟨\Delta Y,\Delta Y{⟩}_{2}Y,$$

$$⟨N,Y{⟩}_2=1,⟨N,N{⟩}_2=0,⟨N,Y_k{⟩}_2=0,⟨Y_i,Y_j{⟩}_2={\delta }_{ij},1\le i,j,k\le n.$$

$${\mathbb{R}}_2^{n+3}=\text{span}\{Y,N\}\oplus \text{span}\{Y_1,\cdots ,Y_n\}\oplus \mathbb{V},$$

$$\begin{array}{rl}& \mathrm{d}Y=\sum _i{\omega }_i{Y}_i,\\ & \mathrm{d}N=\sum _{ij}{A}_{ij}{\omega }_j{Y}_i+\sum _i{C}_i{\omega }_i\xi ,\\ & \mathrm{d}{Y}_i=-\sum _j{A}_{ij}{\omega }_{j}Y-{\omega }_{i}N+\sum _j{\omega }_{ij}{Y}_j+\sum _j{B}_{ij}{\omega }_j\xi ,\\ & \mathrm{d}\xi =\sum _i{C}_i{\omega }_{i}Y+\sum _{ij}{B}_{ij}{\omega }_j{Y}_i,\end{array}$$

$$\sum _j{C}_{i,j}{\omega }_j=d{C}_i+\sum _k{C}_k{\omega }_{kj},$$

$$\sum _k{A}_{ij,k}{\omega }_k=d{A}_{ij}+\sum _k{A}_{ik}{\omega }_{kj}+\sum _k{A}_{kj}{\omega }_{ki},$$

$$\sum _k{B}_{ij,k}{\omega }_k=d{B}_{ij}+\sum _k{B}_{ik}{\omega }_{kj}+\sum _k{B}_{kj}{\omega }_{ki}.$$

$$A_{ij}=A_{ji},B_{ij}=B_{ji},$$

$$A_{ij,k}-A_{ik,j}=B_{ij}{C}_k-B_{ik}{C}_j,$$

$$B_{ij,k}-B_{ik,j}={\delta }_{ij}{C}_k-{\delta }_{ik}{C}_j,$$

$$C_{i,j}-C_{j,i}=\sum _k(B_{ik}{A}_{kj}-B_{jk}{A}_{ki}),$$

$$R_{ijkl}=B_{il}{B}_{jk}-B_{ik}{B}_{jl}+A_{ik}{\delta }_{jl}+A_{jl}{\delta }_{ik}-A_{il}{\delta }_{jk}-A_{jk}{\delta }_{il}.$$

$$\begin{array}{rl}& \text{tr}(A)=\frac{1}{2n}(n^2\kappa -1),R_{ij}=\text{tr}(A){\delta }_{ij}+(n-2)A_{ij}+\sum _k{B}_{ik}{B}_{kj},\\ & (1-n)C_i=\sum _j{B}_{ij,j},\sum _{ij}{B}_{ij}^2=\frac{n-1}{n},\sum _i{B}_{ii}=0,\end{array}$$

$${\Delta }_{M}x=nH{e}_{n+1}.$$

$$y:M^n\to C^{n+2},y=(\frac{<x,x{>}_1+1}{2},x,\frac{<x,x{>}_1-1}{2}).$$

$$⟨\Delta y,\Delta y{⟩}_2-n^2{\kappa }_M=\frac{n}{n-1}(-|II{|}^2+n|H{|}^2)=-e^{2\tau }.$$

$$\begin{array}{rl}& g=\frac{n}{n-1}(|II{|}^2-n|H{|}^2)<\text{d}x,\text{d}x{>}_1:=e^{2\tau }I,Y=e^{\tau }y,\\ & \xi =-Hy+(<x,e_{n+1}{>}_1,e_{n+1},<x,e_{n+1}{>}_1).\end{array}$$

$$\begin{array}{rl}& A_{ij}=e^{-2\tau }[{\tau }_i{\tau }_j-h_{ij}H-{\tau }_{i,j}+\frac{1}{2}(-|\nabla \tau {|}^2+|H{|}^2){\delta }_{ij}],\\ & B_{ij}=e^{-\tau }(h_{ij}-H{\delta }_{ij}),C_i=e^{-2\tau }(H{\tau }_i-H_i-\sum _j{h}_{ij}{\tau }_j),\end{array}$$

$$\begin{array}{rl}& g=\frac{n}{n-1}(|II{|}^2-n|H{|}^2)<\text{d}x,\text{d}x{>}_1:=e^{2\tau }I,\\ & Y=e^{\tau }(1,x)=e^{\tau }y,\xi =-Hy+(0,e_{n+1}).\end{array}$$

$$\begin{array}{rl}& g=\frac{n}{n-1}(|II{|}^2-n|H{|}^2)<\text{d}x,\text{d}x{>}_2:=e^{2\tau }I,\\ & Y=e^{\tau }(x,1)=e^{\tau }y,\xi =-Hy+(e_{n+1},0).\end{array}$$

Full text

2000 Mathematics Subject Classification: 53A30, 53B25.

Key words: Blaschke tnesor, para-Blaschke tensor, para-Blaschke isoparametric hypersurface, conformal isoparametric hypersurface.

1 Introduction

Recently the Möbius geometry of submanifolds in Riemannian space forms has been studied extensively and a lot of interesting results have been obtained. Especially, Many special hypersurfaces were classified under Möbius transformation group (for example, [2],[3],[4],[5],[6],[9],[10],[13],[14]). As its parallel generalization, the conformal geometry of submanifolds in Lorentzian space forms is another important branch of conformal geometry, but there are less results in Lorentzian space forms than in Riemannian space forms. In this paper, we study the para-Blaschke isoparametric spacelike hypersurfaces in Lorentzian space forms.

Let $\mathbb{R}^{n+2}_{s}$ be the real vector space $\mathbb{R}^{n+2}$ with the Lorentzian product $\langle,\rangle_{s}$ given by

[TABLE]

For any $a>0$, the standard sphere $\mathbb{S}^{n+1}(a)$, the hyperbolic space $\mathbb{H}^{n+1}(-a)$, the de sitter space $\mathbb{S}^{n+1}_{1}(a)$ and the anti-de sitter space $\mathbb{H}^{n+1}_{1}(-a)$ are defined by

[TABLE]

Let $M^{n+1}_{1}(c)$ be a Lorentzian space form. When $c=0$, $M^{n+1}_{1}(c)=\mathbb{R}^{n+1}_{1}$. When $c=1$, $M^{n+1}_{1}(c)=\mathbb{S}^{n+1}_{1}(1)$. When $c=-1$, $M^{n+1}_{1}(c)=\mathbb{H}^{n+1}_{1}(-1)$.

For Lorentzian space forms $M^{n+1}_{1}(c)$, there exists a united conformal compactification $\mathbb{Q}^{n+1}_{1}$, which is the projectivized light cone in $\mathbb{R}P^{n+2}$ induced from $\mathbb{R}^{n+3}_{2}$. Using the conformal compactification $\mathbb{Q}^{n+1}_{1}$, we study the conformal geometry of spacelike hypersurfaces in $M_{1}^{n+1}(c)$. We define the conformal metric $g$ and the conformal second fundamental form $B$ on an umbilic-free spacelike hypersurface, which determine the spacelike hypersurface up to a conformal transformation of $M^{n+1}_{1}(c)$. Another two conformal invariants are the conformal $1$-form $C$ and the Blaschke tensor $A$ (see Sect.2).

Since $A$ and $B$ are symmetric $(0,2)$-tensor, their eigenvalues are real. We define two kind of special spacelike hypersurfaces: the conformal isoparametric spacelike hypersurfaces and the Blaschke isoparametric spacelike hypersurfaces. A spacelike hypersurface is called a conformal isoparametric spacelike hypersurface, if it satisfies two conditions: (1) $C=0,$ (2) all the eigenvalues of $B$ are constant. Similarly, we define the Blaschke isoparametric spacelike hypersurface by another symmetric tensor, the Blaschke tensor $A$. The para-Blaschke tensor defined by $D^{\lambda}:=A+\lambda B$ for some constant $\lambda$. Clearly the para-Blaschke tensor is still a symmetric $(0,2)$ tensor, thus its eigenvalues are real. Using the para-Blaschke tensor, we can define similarly the para-Blaschke isoparametric spacelike hypersurface.

Recently, some interesting resultes on the spacelike hypersurfaces with some special conformal invariants are obtained. C. X. Nie et al. classified the spacelike hypersurfaces with parallel conformal second fundamental form in [17], and classified the Blaschke isoparametric spacelike hypersurfaces with two distinct principal curvatures in [16]. X. X. Li et al. classified the spacelike hypersurfaces with parallel Blaschke tensor in [11] and the spacelike hypersurfaces with with parallel para-Blaschke tensor in [12]. T.Z. Li and C.X. Nie classified completely the conformal isoparametric spacelike hypersurfaces in [8]. Clearly if the para-blaschke tensor of a spacelike hypersurface is parallel, then the spacelike hypersurface is para-Blaschke isoparametric. In this paper, we prove that a para-Blaschke isoparametric spacelike hypersurface is a conformal isoparametric spacelike hypersurface provided that the para-Blaschke tensor has more than two distinct eigenvalues. Simultaneously, we classify completely the para-Blaschke isoparametric spacelike hypersurfaces. Our main theorems are as follows.

Theorem 1.1**.**

*Let $x:M^{n}\to M^{n+1}_{1}(c),n\geq 2,$ be an umbilic-free spacelike hypersurface in an $(n+1)$-dimensional Lorentzian space form $M^{n+1}_{1}(c)$. We assume that the conformal $1$-form of $x$ vanishes. Then we have

(1),If the spacelike hypersurface is a conformal isoparametric spacelike hypersurface, then the spacelike hypersurface is also a para-Blaschke isoparametric spacelike hypersurface.

(2),If the spacelike hypersurface is a para-Blaschke isoparametric spacelike hypersurface and the number of the distinct eigenvalues of the para-Blaschke tensor $D^{\lambda}$ is more than two, then the spacelike hypersurface is also a conformal isoparametric spacelike hypersurface.*

Theorem 1.2**.**

*Let $x:M^{n}\to M^{n+1}_{1}(c),n\geq 2,$ be an umbilic-free spacelike hypersurface in an $(n+1)$-dimensional Lorentzian space form $M^{n+1}_{1}(c)$. If the hypersurface is para-Blaschke isoparametric, then $x$ is locally conformal equivalent to one of the following hypersurfaces:

(1), the spacelike hypersurfaces with constant mean curvature and constant scalar curvature in $M^{n+1}_{1}(c)$;

(2), $\mathbb{S}^{k}(\sqrt{a^{2}+1})\times\mathbb{H}^{n-k}(-a)\subset\mathbb{S}^{n+1}_{1}(1),~{}~{}a>0,~{}1\leq k\leq n-1;$

(3), $\mathbb{H}^{k}(-a)\times\mathbb{H}^{n-k}(-\sqrt{1-a^{2}})\subset\mathbb{H}^{n+1}_{1}(-1),~{}~{}0<a<1,~{}1\leq k\leq n-1;$

(4), $\mathbb{H}^{k}(-a)\times\mathbb{R}^{n-k}\subset\mathbb{R}^{n+1}_{1},~{}~{}a>0,~{}0\leq k\leq n-1;$

(5),$x:\mathbb{H}^{q}(-\sqrt{a^{2}-1})\times\mathbb{S}^{p}(a)\times\mathbb{R}^{+}\times\mathbb{R}^{n-p-q-1}\rightarrow\mathbb{R}^{n+1}_{1},$ defined by*

[TABLE]

*where $u^{\prime}\in\mathbb{H}^{q}(-\sqrt{a^{2}-1}),u^{\prime\prime}\in\mathbb{S}^{p}(a),u^{\prime\prime\prime}\in\mathbb{R}^{n-p-q-1},~{}~{}a>1;$

(6) the spacelike hypersurfaces defined by Example 3.5 (see Sect.3);

(7) the spacelike hypersurfaces defined by Example 3.6 (see Sect.3).*

When $\lambda=0$, $D^{\lambda}=A$. Theorem 1.2 implies that the conformal isoparametric spacelike hypersurfaces and the Blaschke isoparametric spacelike hypersurfaces are almost equivalent. Therefore from the results in [8], we have the following results.

Corollary 1.1**.**

Let $x:M^{n}\to M^{n+1}_{1}(c),n\geq 2,$ be an umbilic-free spacelike hypersurface in the $(n+1)$-dimensional Lorentzian space form $M^{n+1}_{1}(c)$ with $r$ distinct eigenvalues of the Blaschke tensor. If the hypersurface is Blaschke isoparametric and $r\geq 3$, then $r=3$ and $x$ is locally conformal equivalent to the following spacelike hypersurface:

[TABLE]

defined by $x(u^{\prime},u^{\prime\prime},t,u^{\prime\prime\prime})=(tu^{\prime},tu^{\prime\prime},u^{\prime\prime\prime}),$ where $u^{\prime}\in\mathbb{H}^{q}(-\sqrt{a^{2}-1}),u^{\prime\prime}\in\mathbb{S}^{p}(a),u^{\prime\prime\prime}\in\mathbb{R}^{n-p-q-1},~{}~{}a>1.$

This paper is organized as follows. In section 2, we study the conformal geometry of spacelike hypersurfaces in $M_{1}^{n+1}(c)$. In section 3, we give some examples of special spacelike hypersurfaces. In section 4 we give the proof of our main theorems.

2 Conformal geometry of spacelike Hypersurfaces

In this section, following Wang’s idea in paper [19], we define some conformal invariants on a spacelike hypersurface and give a congruent theorem of the spacelike hypersurfaces under the conformal group of $M^{n+1}_{1}(c)$.

We denote by $C^{n+2}$ the cone in $\mathbb{R}^{n+3}_{2}$ and by $\mathbb{Q}^{n+1}_{1}$ the conformal compactification space in $\mathbb{R}P^{n+2}$,

[TABLE]

[TABLE]

Let $O(n+3,2)$ be the Lorentzian group of $\mathbb{R}^{n+3}_{2}$ keeping the Lorentzian product $\langle X,Y\rangle_{2}$ invariant. Then $O(n+3,2)$ is a transformation group on $\mathbb{Q}^{n+1}_{1}$ defined by

[TABLE]

Topologically $\mathbb{Q}^{n+1}_{1}$ is identified with the compact space $S^{n}\times S^{1}/S^{0}$, which is endowed by a standard Lorentzian metric $h=g_{S^{n}}\oplus(-g_{S^{1}})$, where $g_{S^{k}}$ denotes the standard metric of the $k$-dimensional sphere $S^{k}$. Then $\mathbb{Q}^{n+1}_{1}$ has conformal metric

[TABLE]

and $[O(n+3,2)]$ is the conformal transformation group of $\mathbb{Q}^{n+1}_{1}$(see[1, 18]).

Denoting $P=\{[X]\in\mathbb{Q}^{n+1}_{1}|x_{1}=x_{n+2}\},~{}~{}P_{-}=\{[X]\in\mathbb{Q}^{n+1}_{1}|x_{n+2}=0\},~{}~{}P_{+}=\{[X]\in\mathbb{Q}^{n+1}_{1}|x_{1}=0\}$, we can define the following conformal diffeomorphisms,

[TABLE]

We may regard $\mathbb{Q}^{n+1}_{1}$ as the common compactification of $\mathbb{R}^{n+1}_{1},\mathbb{S}^{n+1}_{1}(1),\mathbb{H}^{n+1}_{1}(-1)$.

Let $x:M^{n}\rightarrow M^{n+1}_{1}(c)$ be a spacelike hypersurface. Using $\sigma_{c}$, we obtain the hypersurface in $\mathbb{Q}^{n+1}_{1}$, $\sigma_{c}\circ x:M^{n}\rightarrow\mathbb{Q}^{n+1}_{1}$. From [1], we have the following theorem.

Theorem 2.1**.**

Two hypersurfaces $x,\bar{x}:M^{n}\rightarrow M^{n+1}_{1}(c)$ are conformally equivalent if and only if there exists $T\in O(n+3,2)$ such that $\sigma_{c}\circ x=T(\sigma_{c}\circ\bar{x}):M^{n}\rightarrow\mathbb{Q}^{n+1}_{1}$.

Since $x:M^{n}\rightarrow M^{n+1}_{1}(c)$ is a spacelike hypersurface, $(\sigma_{c}\circ x)_{*}(TM^{n})$ is a positive definite subbundle of $T{\mathbb{Q}}^{n+1}_{1}$. For any local lift $Z$ of the standard projection $\pi:C^{n+2}\rightarrow\mathbb{Q}^{n+1}_{1}$, we get a local lift $y=Z\circ\sigma_{c}\circ x:U\rightarrow C^{n+1}$ of $\sigma_{c}\circ x:M\rightarrow{\mathbb{Q}}^{n+1}_{1}$ in an open subset $U$ of $M^{n}$. Thus $\langle\text{d}y,\text{d}y\rangle_{2}=\rho^{2}\langle dx,dx\rangle_{s}$ is a local metric, where $\rho\in C^{\infty}(U)$. We denote by $\Delta$ and $\kappa$ the Laplacian operator and the normalized scalar curvature with respect to the local positive definite metric $\langle\text{d}y,\text{d}y\rangle$, respectively. Similar to Wang’s proof of Theorem 1.2 in [19], we can get the following theorem.

Theorem 2.2**.**

Let $x:M^{n}\rightarrow M^{n+1}_{1}(c)$ be a spacelike hypersurface, then the 2-form $g=-(\langle\Delta y,\Delta y\rangle_{2}-n^{2}\kappa)\langle\text{d}y,\text{d}y\rangle_{2}$ is a globally defined conformal invariant. Moreover, $g$ is positive definite at any non-umbilical point of $M^{n}$.

We call $g$ the conformal metric of the spacelike hypersurface $M^{n}$. There exists a unique lift

[TABLE]

such that $g=\langle\text{d}Y,\text{d}Y\rangle_{2}$. We call $Y$ the conformal position vector of the spacelike hypersurface $M^{n}$. Theorem 2.2 implies that

Theorem 2.3**.**

Two spacelike hypersurfaces $x,\bar{x}:M^{n}\rightarrow M^{n+1}_{1}(c)$ are conformally equivalent if and only if there exists $T\in O(n+3,2)$ such that $\bar{Y}=YT$, where $Y,\tilde{Y}$ are the conformal position vector of $x,\tilde{x}$, respectively.

Let $\{E_{1},\cdots,E_{n}\}$ be a local orthonormal basis of $M^{n}$ with respect to $g$ with dual basis $\{\omega_{1},\cdots,\omega_{n}\}$. Denote $Y_{i}=E_{i}(Y)$ and define

[TABLE]

where $\Delta$ is the Laplace operator of $g$, then we have

[TABLE]

We may decompose $\mathbb{R}^{n+3}_{2}$ such that

[TABLE]

where $\mathbb{V}\bot\text{span}\{Y,N,Y_{1},\cdots,Y_{n}\}$. We call $\mathbb{V}$ the conformal normal bundle of $x$, which is linear bundle. Let $\xi$ be a local section of $\mathbb{V}$ and $<\xi,\xi>_{2}=-1$, then $\{Y,N,Y_{1},\cdots,Y_{n},\xi\}$ forms a moving frame in $\mathbb{R}^{n+3}_{2}$ along $M^{n}$. We write the structure equations as follows,

[TABLE]

where $\omega_{ij}(=-\omega_{ij})$ are the connection 1-forms on $M^{n}$ with respect to $\{\omega_{1},\cdots,\omega_{n}\}$. It is clear that $A=\sum_{ij}A_{ij}\omega_{j}\otimes\omega_{i},~{}B=\sum_{ij}B_{ij}\omega_{j}\otimes\omega_{i},~{}C=\sum_{i}C_{i}\omega_{i}$ are globally defined conformal invariants. We call $A,~{}B$ and $C$ the Blaschke tensor, the conformal second fundamental form and the conformal $1$-form, respectively. The covariant derivatives of these tensors with respect to $\omega_{ij}$ are defined by:

[TABLE]

[TABLE]

[TABLE]

By exterior differentiation of structure equations (2.1), we can get the integrable conditions of the structure equations

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Furthermore, we have

[TABLE]

where $\kappa$ is the normalized scalar curvature of $g$. From (2.6), we see that when $n\geq 3$, all coefficients in the structure equations are determined by the conformal metric $g$ and the conformal second fundamental form $B$, thus we get the following conformal congruent theorem.

Theorem 2.4**.**

Two spacelike hypersurfaces $x,\bar{x}:M^{n}\rightarrow M^{n+1}_{1}(c)(n\geq 3)$ are conformally equivalent if and only if there exists a diffeomorphism $\varphi:M^{n}\rightarrow M^{n}$ which preserves the conformal metric and the conformal second fundamental form.

Next we give the relations between the conformal invariants and the isometric invariants of a spacelike hypersurface in $M^{n+1}_{1}(c)$.

First we consider the spacelike hypersurface $x:M^{n}\rightarrow\mathbb{R}^{n+1}_{1}$ in $\mathbb{R}^{n+1}_{1}$. Let $\{e_{1},\cdots,e_{n}\}$ be an orthonormal local basis with respect to the induced metric $I=<dx,dx>_{1}$ with dual basis $\{\theta_{1},\cdots,\theta_{n}\}$. Let $e_{n+1}$ be a normal vector field of $x$ , $<e_{n+1},e_{n+1}>_{1}=-1$. Let $II=\sum_{ij}h_{ij}\theta_{i}\otimes\theta_{j}$ denote the second fundamental form, the mean curvature $H=\frac{1}{n}\sum_{i}h_{ii}$. Denote by $\Delta_{M}$ the Laplacian operator and $\kappa_{M}$ the normalized scalar curvature for $I$. By structure equation of $x:M^{n}\rightarrow\mathbb{R}^{n+1}_{1}$ we get that

[TABLE]

There is a local lift of $x$

[TABLE]

It follows from (2.7) that

[TABLE]

Therefore the conformal metric $g$, conformal position vector of $x$ and $\xi$ have the following expression,

[TABLE]

By a direct calculation we get the following expression of the conformal invariants,

[TABLE]

where $\tau_{i}=e_{i}(\tau)$ and $|\nabla\tau|^{2}=\sum_{i}\tau_{i}^{2}$, and $\tau_{i,j}$ is the Hessian of $\tau$ for $I$ and $H_{i}=e_{i}(H)$.

For a spacelike hypersurface $x:M^{n}\rightarrow\mathbb{S}^{n+1}_{1}(1)$, the conformal metric $g$, conformal position vector of $x$ and $\xi$ have the following expression,

[TABLE]

For a spacelike hypersurface $x:M^{n}\rightarrow\mathbb{H}^{n+1}_{1}(-1)$, the conformal metric $g$, conformal position vector of $x$ and $\xi$ have the following expression,

[TABLE]

Using the similar calculation from (2.10) and (2.11), we have the following united expression of the conformal invariants,

[TABLE]

where $c=1$ for $x:M^{n}\rightarrow S^{n+1}_{1}(1)$, and $c=-1$ for $x:M^{n}\rightarrow H^{n+1}_{1}(-1)$.

3 Typical examples

In this section, we present some examples of the spacelike hypersurfaces in $M^{n+1}_{1}(c)$ with constant eigenvalues of para-Blaschke tensor.

Example 3.1**.**

For constant $a>0$, let $x_{1}:\mathbb{H}^{k}(-1)\to\mathbb{R}^{k+1}_{1}$ be the standard embedding and $y:\mathbb{R}^{n-k}\to\mathbb{R}^{n-k}$ identity. We define the spacelike hypersurface

[TABLE]

Let $\xi=(\frac{1}{a}x_{1},\overrightarrow{0})$ be the normal vector field of $x$. Thus

[TABLE]

where $I_{\mathbb{H}^{k}(-a)}$ denotes the standard metric on $\mathbb{H}^{k}(-a)$ and $I_{\mathbb{R}^{n-k}}$ the standard metric on $\mathbb{R}^{n-k}$.

Let $\{e_{1},\cdots,e_{k}\}$ be a local fields of orthonormal basis on $\mathbb{H}^{k}(-a)$ and $\{e_{k+1},\cdots,e_{n}\}$ a local fields of orthonormal basis on $\mathbb{R}^{n-k}$, then $\{e_{1},\cdots,e_{n}\}$ is a local fields of orthonormal basis on $\mathbb{H}^{k}(-a)\times\mathbb{R}^{n-k}$. Thus, under the local fields of orthonormal basis $\{e_{1},\cdots,e_{n}\}$,

[TABLE]

Under the local fields of orthonormal basis, from (2.9), we have

[TABLE]

where

[TABLE]

Thus $(D^{\lambda}_{ij})=diag(\underbrace{d_{1},\cdots,d_{1}}_{k},\underbrace{d_{2},\cdots,d_{2}}_{n-k})$ and $d_{1}=a_{1}+\lambda b_{1},~{}d_{2}=a_{2}+\lambda b_{2}$.

Example 3.2**.**

Let $x_{1}:\mathbb{S}^{k}(1)\to\mathbb{R}^{k+1}$ and $x_{2}:\mathbb{H}^{n-k}(-1)\to\mathbb{R}^{n-k+1}_{1}$ be two standard embedings. For constant $a>0$, we define the spacelike hypersurface

[TABLE]

Let $\xi=(ax_{1},\sqrt{1+a^{2}}x_{2})$ be the normal vector field of $x$. Thus

[TABLE]

[TABLE]

Let $\{e_{1},\cdots,e_{k}\}$ be a local fields of orthonormal basis on $\mathbb{S}^{k}(\sqrt{1+a^{2}})$ and $\{e_{k+1},\cdots,e_{n}\}$ a local fields of orthonormal basis on $\mathbb{H}^{n-k}(-a)$, then $\{e_{1},\cdots,e_{n}\}$ is a local fields of orthonormal basis on $\mathbb{S}^{k}(\sqrt{1+a^{2}})\times\mathbb{H}^{n-k}(-a)$. Thus, under the local fields of orthonormal tangent frame $\{e_{1},\cdots,e_{n}\}$,

[TABLE]

Under the local fields of orthonormal basis, from (2.11), we have

[TABLE]

where

[TABLE]

Thus $(D^{\lambda}_{ij})=diag(\underbrace{d_{1},\cdots,d_{1}}_{k},\underbrace{d_{2},\cdots,d_{2}}_{n-k})$ and $d_{1}=a_{1}+\lambda b_{1},~{}d_{2}=a_{2}+\lambda b_{2}$.

Example 3.3**.**

Let $x_{1}:\mathbb{H}^{k}(-1)\to\mathbb{R}^{k+1}_{1}$ and $x_{2}:\mathbb{H}^{n-k}(-1)\to\mathbb{R}^{n-k+1}_{1}$ be two standard embedings. For constant $a$ satisfying $0<a<1$, We we define the spacelike hypersurface

[TABLE]

Let $\xi=(-ax_{1},\sqrt{1-a^{2}}x_{2})$ be the normal vector field of $x$. Thus

[TABLE]

[TABLE]

Let $\{e_{1},\cdots,e_{k}\}$ be a local fields of orthonormal basis on $\mathbb{H}^{k}(-a)$ and $\{e_{k+1},\cdots,e_{n}\}$ a local fields of orthonormal basis on $\mathbb{H}^{n-k}(-\sqrt{1-a^{2}})$, then $\{e_{1},\cdots,e_{n}\}$ is a local fields of orthonormal basis on $\mathbb{H}^{k}(-a)\times\mathbb{H}^{n-k}(-\sqrt{1-a^{2}})$. Thus, under the local fields of orthonormal basis $\{e_{1},\cdots,e_{n}\}$,

[TABLE]

Under the local fields of orthonormal basis, from (2.11), we have

[TABLE]

where

[TABLE]

Thus $(D^{\lambda}_{ij})=diag(\underbrace{d_{1},\cdots,d_{1}}_{k},\underbrace{d_{2},\cdots,d_{2}}_{n-k})$ and $d_{1}=a_{1}+\lambda b_{1},~{}d_{2}=a_{2}+\lambda b_{2}$.

Example 3.4**.**

Let $p,q$ be any two given natural numbers with $p+q<n$ and a real number $a>1$. We define the spacelike hypersurface

[TABLE]

defined by

[TABLE]

where $u^{\prime}\in\mathbb{H}^{q}(-\sqrt{a^{2}-1}),u^{\prime\prime}\in\mathbb{S}^{p}(a),u^{\prime\prime\prime}\in\mathbb{R}^{n-p-q-1}.$

Let $b=\sqrt{a^{2}-1}$. One of the normal vector of $x$ can be taken as

[TABLE]

The first and second fundamental form of $x$ are given by

[TABLE]

[TABLE]

Thus the mean curvature of $x$ satisfies

[TABLE]

and $e^{2\tau}=\frac{n}{n-1}[\sum_{ij}h^{2}_{ij}-nH^{2}]=\frac{p(n-p)b^{4}-2pqa^{2}b^{2}+q(n-q)a^{4}}{(n-1)t^{2}}:=\frac{\alpha^{2}}{t^{2}}.$

From (2.8) and (2.12), we see that the conformal 1-form $C=0$, and the conformal metric and the conformal second fundamental form of $x$ are given by

[TABLE]

where $b_{1}=\frac{pb^{2}-(n-q)a^{2}}{nab\alpha},~{}b_{2}=\frac{qa^{2}-(n-p)b^{2}}{nab\alpha},~{}b_{3}=\frac{pb^{2}+qa^{2}}{nab\alpha},$ and

[TABLE]

[TABLE]

Thus $(D^{\lambda}_{ij})=diag(\underbrace{d_{1},\cdots,d_{1}}_{q},\underbrace{d_{2},\cdots,d_{2}}_{p},\underbrace{d_{3},\cdots,d_{3}}_{n-p-q})$, $d_{i}=a_{i}+\lambda b_{i},~{}~{}i=1,2,3.$

Example 3.5**.**

Given constants $\lambda,r(r>0)$, we define the spacelike hypersurface

[TABLE]

Here $y=(y_{0},y_{2}):\mathbb{H}^{n-k}(-r)\to\mathbb{R}^{n-k+1}_{1}$ is a standard embedding, and $y_{1}:M^{k}\to\mathbb{S}^{k+1}_{1}(r)\subset\mathbb{R}^{k+2}_{1}$ is a umbilic-free spacelike hypersurface with constant scalar curvature $R_{1}$ and the mean curvature $H_{1}$ satisfying $R_{1}=\frac{nk(k-1)+(n-1)r^{2}}{nr^{2}}-n(n-1)\lambda^{2},~{}~{}~{}H_{1}=\frac{n}{k}\lambda,$ respectively.

Using the structure of the spacelike hypersurface $y_{1}$, we have

[TABLE]

where $\triangle_{1}$ is the Laplacian with respect to the first fundamental form $<dy_{1},dy_{1}>_{1}$ and $\xi_{1}$ is the unit normal vector field of $y_{1}$.

The standard embedding $y=(y_{0},y_{2}):\mathbb{H}^{n-k}(-r)\to\mathbb{R}^{n-k+1}_{1}$ is totally umbilical, thus the scalar curvature $R_{2}=-\frac{(n-k)(n-k-1)}{r^{2}},$ and

[TABLE]

where $\triangle_{2}$ is the Laplacian with respect to the first fundamental form $<dy,dy>_{1}$.

The conformal position vector of the spacelike hypersurface is

[TABLE]

Since the conformal metric $g=<dY,dY>$,

[TABLE]

We take a local orthonormal basis $\{e_{p},p=1,...,k\}$ on $TM^{k}$, and $\{e_{q},q=k+1,...,n\}$ on $T\mathbb{H}^{n-k}(-r)$. Thus $\{e_{1},\cdots,e_{k},e_{k+1},\cdots,e_{n}\}$ is a local orthonormal basis on $T(M^{k}\times\mathbb{H}^{n-k}(-r))$ and

[TABLE]

Under the local basis $\{e_{1},\cdots,e_{n}\}$, using $A_{ij}=<Y_{i},N_{j}>_{2},B_{ij}=<Y_{i},\xi_{j}>_{2},$ and $D_{ij}=A_{ij}+\lambda B_{ij}$, we have $C=0$ and

[TABLE]

Example 3.6**.**

Given constants $\lambda,r(r>0)$, let

[TABLE]

be a spacelike hypersurface with constant scalar curvature $R_{1}$ and the mean curvature $H_{1}$ satisfying $R_{1}=\frac{-nk(k-1)+(n-1)r^{2}}{nr^{2}}-n(n-1)\lambda^{2},~{}H_{1}=\frac{n}{k}\lambda$, respectively.

Since $-y_{0}^{2}-\tilde{y}_{0}^{2}+<y_{1},y_{1}>=-r^{2}$, $y_{0}$ and $\tilde{y}_{0}$ can not be zero simultaneously. Without loss of generality, we assume that $y_{0}\neq 0$. In this case, we define the spacelike hypersurface

[TABLE]

where $y_{2}:\mathbb{S}^{n-k}(r)\to\mathbb{R}^{n-k+1}$ is a round sphere with radius $r$.

Using the structure equation of the spacelike hypersurface $y$, we have

[TABLE]

where $\triangle_{1}$ is the Laplacian with respect to the first fundamental form $<dy,dy>_{1}$ and $\xi_{1}$ is the unit normal vector field of $y$.

The round sphere $y_{2}:\mathbb{S}^{n-k}(r)\rightarrow R^{n-k+1}$ is totally umbilical, thus the scalar curvature is $R_{2}=\frac{(n-k)(n-k-1)}{r^{2}},$ and

[TABLE]

where $\triangle_{2}$ is the Laplacian with respect to the first fundamental form $<dy_{2},dy_{2}>$.

The conformal position vector of the spacelike hypersurface is

[TABLE]

Since the conformal metric $g=<dY,dY>$, we have

[TABLE]

We take a local orthonormal basis $\{e_{p},p=1,...,k\}$ on $TM^{k}$, and $\{e_{q},q=k+1,...,n\}$ on $T\mathbb{S}^{n-k}(r)$. Thus $\{e_{1},\cdots,e_{k},e_{k+1},\cdots,e_{n}\}$ is a local orthonormal basis on $T(M^{k}\times\mathbb{S}^{n-k}(r))$ and

[TABLE]

Under the local basis $\{e_{1},\cdots,e_{n}\}$, using $A_{ij}=<Y_{i},N_{j}>_{2},B_{ij}=<Y_{i},\xi_{j}>_{2},$ we have $C=0$ and

[TABLE]

In [8], authors classified completely the conformal isoparametric spacelike hypersurfaces in $M^{n+1}_{1}(c)$.

Theorem 3.1**.**

[8*]** Let $x:M^{n}\to M^{n+1}_{1}(c)$ be a spacelike hypersurface in $M_{1}^{n+1}(c)$ with two distinct principal curvatures. If the conformal form vanishes, then locally $x$ is conformally equivalent to one of the following hypersurfaces

(1), $\mathbb{S}^{k}(\sqrt{a^{2}+1})\times\mathbb{H}^{n-k}(-a)\subset\mathbb{S}^{n+1}_{1}(1),~{}~{}a>0,~{}1\leq k\leq n-1;$

(2), $\mathbb{H}^{k}(-a)\times\mathbb{H}^{n-k}(-\sqrt{1-a^{2}})\subset\mathbb{H}^{n+1}_{1}(-1),~{}~{}0<a<1,~{}1\leq k\leq n-1;$

(3), $\mathbb{H}^{k}(-a)\times\mathbb{R}^{n-k}\subset\mathbb{R}^{n+1}_{1},~{}~{}a>0,~{}1\leq k\leq n-1.$*

Theorem 3.2**.**

[8]** Let $x:M^{n}\to M^{n+1}_{1}(c)$ be a conformal isoparametric spacelike hypersurface in $M_{1}^{n+1}(c)$ with $r$ distinct principal curvatures. If $r\geq 3$, then $r=3$, and locally $x$ is conformally equivalent to the following hypersurface

[TABLE]

defined by

[TABLE]

where $u^{\prime}\in\mathbb{H}^{q}(-\sqrt{a^{2}-1}),u^{\prime\prime}\in\mathbb{S}^{p}(a),u^{\prime\prime\prime}\in\mathbb{R}^{n-p-q-1},~{}~{}a>1.$

The following theorem is need in the proof of the main theorem, readers refer [7].

Theorem 3.3**.**

[7]** Let $x:M^{n}\rightarrow M^{n+1}_{1}(c)$ be a spacelike hypersurface without umbilical points. If conformal invariants of $x$ satisfy

[TABLE]

Then $x$ is conformally equivalent to a spacelike hypersurface with constant mean curvature and constant scalar curvature.

4 Proof of the main Theorem

Proof of Theorem 1.1. From Theorem 3.1, Theorem 3.2, Example 3.1, Example 3.2, Example 3.3 and Example 3.4, we know that if the spacelike hypersurface is conformally isoparametric, then the spacelike hypersurface is also para-Blaschke isoparametric.

Next we assume that the spacelike hypersurface is a para-Blaschke isoparametric spacelike hypersurface and the number of the distinct eigenvalues of the para-Blaschke tensor $D^{\lambda}$ is more than two. Since the conformal $1$-form vanishes, we can have a local local orthonormal basis $\{E_{1},\cdots,E_{n}\}$ such that

[TABLE]

Using the covariant derivative $dD^{\lambda}_{ij}+\sum_{k}D^{\lambda}_{kj}\omega_{ki}+\sum_{k}D^{\lambda}_{ik}\omega_{kj}=\sum_{k}D^{\lambda}_{ij,k}\omega_{k}$, we have

[TABLE]

For each $i$ fixed, we define the index set $[i]=\{m|d_{m}=d_{i}\}.$ We have the following results

[TABLE]

The second covariant derivative of $D^{\lambda}_{ij}$ is defined by

[TABLE]

Let $[i]\neq[j]$, we have

[TABLE]

Using the Ricci identities $D^{\lambda}_{ij,ij}-D^{\lambda}_{ij,ji}=\sum_{m}(D^{\lambda}_{mj}R_{miij}+D^{\lambda}_{im}R_{mjij}),$ we get

[TABLE]

For the conformal second fundamental form $B$, we have

[TABLE]

Using (4.17), we get

[TABLE]

Let $k=i$ in (4.19), and using $D^{\lambda}_{ij,i}=0$ we obtain

[TABLE]

In order to prove that $b_{i}$ is a constant, we only need to prove

[TABLE]

For each $i$ fixed, we consider two cases:

Case 1. There exist $j,\ k$ such that

[TABLE]

Case 2. For all $j,\ k$, we have $D^{\lambda}_{ij,k}=0$.

Now we consider Case 1, since

[TABLE]

from (4.19), we get

[TABLE]

Thus

[TABLE]

From (4.20), since $E_{l}(b_{k})=E_{l}(b_{j})=0,\ \ l\in[i]$, we have

[TABLE]

For Case 2. Since $D^{\lambda}_{ij,k}=0,\ \forall j,k$, from (4.18), we get

[TABLE]

From (2.5) we have $0=R_{ijij}=-b_{i}b_{j}+a_{i}+a_{j},\ j\notin[i].$ Since $a_{i}=d_{i}-\lambda b_{i}$, we have

[TABLE]

Note that the number of the distinct eigenvalues of the para-Blaschke tensor $D^{\lambda}$ is more than two, we can take $j,k$ such that $[j]\neq[i],$ $[k]\neq[i]$ and $[k]\neq[j]$, and

[TABLE]

Thus

[TABLE]

namely

[TABLE]

Noting $E_{l}(b_{j})=E_{l}(b_{k})=0,\ \ l\in[i],\ j,k\notin[i],$ we obtain

[TABLE]

From (4.20), (4.21) and (4.22), it concludes that

[TABLE]

Thus $\{b_{i}|i=1,\cdots,n\}$ are constant and $x$ is conformal isoparametric spacelike hypersurface. Thus we complete the proof of Theorem 1.1.

Next we divide Theorem 1.2 into three cases. If the number of the distinct eigenvalues of the para-Blaschke tensor is $1$, using Theorem 3.3, then we have the following proposition.

Proposition 4.1**.**

Let $x:M^{n}\rightarrow M^{n+1}_{1}(c)$ be a para-Blaschke isoparametric spacelike hypersurface with $r$ distinct eigenvalues of the para-Blaschke tensor. If $r=1$, then $x$ is conformally equivalent to a spacelike hypersurface with constant mean curvature and constant scalar curvature in $M^{n+1}_{1}(c)$.

If the number of the distinct eigenvalues of the para-Blaschke tensor is more than two, then we have the following Proposition by Theorem 3.2 and Theorem 1.1.

Proposition 4.2**.**

Let $x:M^{n}\rightarrow M^{n+1}_{1}(c)$ be a para-Blaschke isoparametric spacelike hypersurface with $r$ distinct eigenvalues of the para-Blaschke tensor. If $r\geq 3$, then $r=3$, and locally $x$ is conformally equivalent to the following hypersurface,

[TABLE]

defined by $x(u^{\prime},u^{\prime\prime},t,u^{\prime\prime\prime})=(tu^{\prime},tu^{\prime\prime},u^{\prime\prime\prime}),$ where $u^{\prime}\in\mathbb{H}^{q}(-\sqrt{a^{2}-1}),u^{\prime\prime}\in\mathbb{S}^{p}(a),u^{\prime\prime\prime}\in\mathbb{R}^{n-p-q-1},~{}~{}a>1.$

Next we assume that the number of the distinct eigenvalues of the para-Blaschke tensor is two, we have

Proposition 4.3**.**

*Let $x:M^{n}\rightarrow M^{n+1}_{1}(c)$ be a para-Blaschke isoparametric spacelike hypersurface with two distinct eigenvalues of the para-Blaschke tensor. Then $x$ is locally conformal equivalent to one of the following hypersurfaces:

(1), $\mathbb{S}^{k}(\sqrt{a^{2}+1})\times\mathbb{H}^{n-k}(-a)\subset\mathbb{S}^{n+1}_{1}(1),~{}~{}a>0,~{}1\leq k\leq n-1;$

(2), $\mathbb{H}^{k}(-a)\times\mathbb{H}^{n-k}(-\sqrt{1-a^{2}})\subset\mathbb{H}^{n+1}_{1}(-1),~{}~{}0<a<1,~{}1\leq k\leq n-1;$

(3), $\mathbb{H}^{k}(-a)\times\mathbb{R}^{n-k}\subset\mathbb{R}^{n+1}_{1},~{}~{}a>0,~{}0\leq k\leq n-1;$

(4) the spacelike hypersurfaces defined by Example 3.5;

(5) the spacelike hypersurfaces defined by Example 3.6.*

Proof.

Since the conformal $1$-form vanishes, we can get a local local orthonormal basis $\{E_{1},\cdots,E_{n}\}$ such that

[TABLE]

Furthermore, we assume that

[TABLE]

Making the following convention on the ranges of indices:

[TABLE]

From (4.16), for all $i,j,p,q,\alpha,\beta$ we have

[TABLE]

Since $\mu$ and $\nu$ are constant, using the total symmetry of $D_{ij,k}$, we can get that $D^{\lambda}$ is parallel, i.e., $D_{ij,k}=0,\forall i,j,k.$

Let $V_{1}$ and $V_{2}$ be the eigen-subbundles of the tangent bundle $TM$ corresponding to $\mu,\ \nu$, respectively. Then

[TABLE]

Since $D^{\lambda}$ is parallel, we have

[TABLE]

which implies that the Riemannian manifold $(M^{n},g)$ can be decomposed locally into a direct product of two Riemannian manifolds $(M_{1},g_{1})$ and $(M_{2},g_{2})$, that is

[TABLE]

Thus $R_{p\alpha p\alpha}=0,$ and from (2.5) we know that

[TABLE]

Claim 1: The eigenvalues of the conformal second fundamental form $B$ satisfy either $b_{1}=b_{2}=...=b_{m_{1}}$, or $b_{m_{1}+1}=...=b_{n}$.

Proof of Claim 1: We assume that $n\geq 3$. From (4.25), we have

[TABLE]

that is

[TABLE]

If $\lambda^{2}+\mu+\nu\neq 0$, (4.26) implies $b_{p}+\lambda\neq 0,\ b_{\alpha}+\lambda\neq 0,\ \forall p,\alpha.$ Since $m_{1}\geq 2$, we have $-(b_{q}+\lambda)(b_{\alpha}+\lambda)+\lambda^{2}+\mu+\nu=0,$ which implies that

[TABLE]

Thus

[TABLE]

Similarly, if $n-m_{1}\geq 2$, we can obtain $b_{m_{1}+1}=b_{m_{1}+2}=...=b_{n}$. Thus the conformal second fundamental form $(B_{ij})$ has two distinct constant eigenvalues.

If $\lambda^{2}+\mu+\nu=0$, (4.26) implies $(b_{p}+\lambda)(b_{\alpha}+\lambda)=0$, which proves the Claim 1.

By Claim 1, the proof of the Proposition 4.3 is divided into the following two cases.

Case I. The conformal second fundamental form $B$ has only two distinct eigenvalues. According to Theorem 3.1, $x$ is locally conformal equivalent to one of the hypersurfaces in Example 3.1, Example 3.2 and Example 3.3.

Case II. The conformal second fundamental form $B$ has more than two distinct eigenvalues.

By Claim 1, we see that either $b_{1}=b_{2}=...=b_{m_{1}}$, or $b_{m_{1}+1}=b_{m_{1}+2}=...=b_{n}$. Without loss of generality, assuming $b_{m_{1}+1}=b_{m_{1}+2}=...=b_{n}$. From the proof of the Claim 1, we know that $b_{m_{1}+1}=b_{m_{1}+2}=...=b_{n}=-\lambda$, $m_{1}\geq 2$ and $\lambda^{2}+\mu+\nu=0$.

Let $\tilde{h}_{pq}=B_{pq}+\lambda\delta_{pq}$, then from (2.5), we obtain the components of curvature tensor on $(M_{1},g_{1})$

[TABLE]

The components of curvature tensor on $(M_{2},g_{2})$

[TABLE]

Hence if $n-m_{1}\geq 2$, $(M_{2},g_{2})$ is of constant sectional curvature $2\nu+\lambda^{2}$.

Since $(2\mu+\lambda^{2})+(2\nu+\lambda^{2})=2(\lambda^{2}+\mu+\nu)=0$ and $\mu\neq\nu$, we need to consider the following two subcases:

Subcase 2.1. $2\mu+\lambda^{2}>0,\ 2\nu+\lambda^{2}<0.$

Set $r=(2\mu+\lambda^{2})^{-\frac{1}{2}}$, then $2\nu+\lambda^{2}=-r^{-2},$ and $(M_{2},g_{2})$ can be locally identified with $\mathbb{H}^{n-m_{1}}(-r)$. Let $v:\mathbb{H}^{n-m_{1}}(-r)\rightarrow\mathbb{R}_{1}^{n-m_{1}+1}$ be the standard totally umbilical hypersurface.

Writing $h^{1}=\sum_{p,q=1}^{m_{1}}\tilde{h}_{pq}\omega_{p}\otimes\omega_{q}$, by $C=0$ and (2.3), we know $h^{1}$ is a Codazzi tensor on $(M_{1},g_{1})$, (4.27) means that there exists a space-like hypersurface

[TABLE]

with $h^{1}$ as its second fundamental form. Clearly, $u$ has at least two non-zero principal curvatures. According to (4.27) and (2.6), we can prove directly that $u$ is of constant mean curvature $H_{1}$ and constant scalar curvature $R_{1}$ satisfying

[TABLE]

Thus $x$ is locally conformal equivalent to the hypersurfaces in Example 3.5.

Subcase 2.2. $2\mu+\lambda^{2}<0,\ 2\nu+\lambda^{2}>0.$

Set $r=(2\nu+\lambda^{2})^{-\frac{1}{2}}$, then $2\mu+\lambda^{2}=-r^{-2},$ and $(M_{2},g_{2})$ can be locally identified with $\mathbb{S}^{n-m_{1}}(r)$. Let $v:\mathbb{S}^{n-m_{1}}(r)\rightarrow\mathbb{R}^{n-m_{1}+1}$ be the standard totally umbilical hypersurface.

Writing $h^{1}=\sum_{pq=1}^{m_{1}}\tilde{h}_{pq}\omega_{p}\otimes\omega_{q}$, by $C=0$ and (2.3), we know $h^{1}$ is a Codazzi tensor on $(M_{1},g_{1})$, (4.27) means that there exists a space-like hypersurface

[TABLE]

with $h^{1}$ as its second fundamental form. Clearly, $u$ has at least two non-zero principal curvatures. According to (4.27) and (2.6), we can prove directly that $u$ is of constant mean curvature $H_{1}$ and constant scalar curvature $R_{1}$ satisfying

[TABLE]

Thus $x$ is locally conformal equivalent to the hypersurfaces in Example 3.6.

Thus we complete the proof of Proposition 4.3 ∎

Using Proposition 4.1, Proposition 4.2 and Proposition 4.3, we finish the proof of Theorem 1.2.

Acknowledgements: Authors are supported by the grant No. 11571037 and No. 11471021 of NSFC.